Concept explainers
Where is the greatest integer function f(x) = [[ x ]] not differentiable? Find a formula for[' and sketch its graph.
To find: The points where the greatest integer function
differentiable and then to find the formula for
Answer to Problem 52E
The greatest integer function is not differentiable at
The formula for the derivative function is
Explanation of Solution
Result Used:
The greatest integer function is defined as follows,
Calculation:
Obtain the points where the greatest integer function
The derivative of the greatest integer function is calculated as follows,
Case 1: x is not an integer.
Since x is not an integer,
Thus,
Case 2: x is an integer.
The left hand derivative of the function is computed as follows,
Since h approaches 0 from left,
This implies that,
Thus, the left hand derivative of the function does not exist. This follows that,
Therefore, the greatest integer function is not differentiable at the integer points.
By case (1) and (2), it is defined that the formula for the derivative is
Graph:
Use the above information and trace the graph of
From the Figure 1, it is clear that the greatest integer function is discontinuous at integer points.
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